Nonlinear stability analysis for thermal convection in the coupled Navier-Stokes-Darcy system

In superposed fluid-porous medium systems, the ratio of the fluid height to the porous medium height exerts a significant influence on the behavior of the coupled system, most notably with its impact on stability and resulting convection cells. Altering the depth ratio slightly can shift convection cells from existing solely in the fluid region to encapsulating the entirety of the fluid and porous regions. With current interest surrounding superposed fluid-porous medium systems in numerous projects of industrial, environmental, and geophysical importance (oil recovery, carbon dioxide sequestration, contamination in sub-soil reservoirs, etc.), being able to predict the critical depth ratio where the shift of convection cells occurs is particularly timely. An investigation through the lens of linear and nonlinear stability analyses provides a more holistic understanding of the nature of the coupled fluid-porous medium system.

In this work, we conduct a novel nonlinear stability analysis for thermal convection in the coupled Navier-Stokes-Darcy system. Armed with this analysis, we explore the linear and nonlinear marginal stability curves in addition to examining the effect certain parameters have on stability, with an emphasis on the small Darcy number limit.